from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1519, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([32,28]))
pari: [g,chi] = znchar(Mod(625,1519))
Basic properties
| Modulus: | \(1519\) | |
| Conductor: | \(1519\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1519.bh
\(\chi_{1519}(25,\cdot)\) \(\chi_{1519}(191,\cdot)\) \(\chi_{1519}(242,\cdot)\) \(\chi_{1519}(408,\cdot)\) \(\chi_{1519}(625,\cdot)\) \(\chi_{1519}(676,\cdot)\) \(\chi_{1519}(842,\cdot)\) \(\chi_{1519}(893,\cdot)\) \(\chi_{1519}(1110,\cdot)\) \(\chi_{1519}(1276,\cdot)\) \(\chi_{1519}(1327,\cdot)\) \(\chi_{1519}(1493,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | 21.21.98353367498957471525704156941061377491148627676882129.1 |
Values on generators
\((1179,344)\) → \((e\left(\frac{16}{21}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1519 }(625, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) |
sage: chi.jacobi_sum(n)